Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{6r^2 + 36r - 42}{6r^2 + 54r + 84}$
First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {6(r^2 + 6r - 7)} {6(r^2 + 9r + 14)} $ $ n = \dfrac{6}{6} \cdot \dfrac{r^2 + 6r - 7}{r^2 + 9r + 14} $ Simplify: $ n = \dfrac{r^2 + 6r - 7}{r^2 + 9r + 14}$ Next factor the numerator and denominator. $ n = \dfrac{(r + 7)(r - 1)}{(r + 7)(r + 2)}$ Assuming $r \neq -7$ , we can cancel the $r + 7$ $ n = \dfrac{r - 1}{r + 2}$ Therefore: $ n = \dfrac{ r - 1 }{ r + 2 }$, $r \neq -7$